Page:A Course of Modern Analysis - 3rd edition - 1920.pdf/17

 CHAPTER I

COMPLEX NUMBERS

11. Rational numbers.

The idea of a set of numbers is derived in the first instance from the consideration of the set of positive* integral numbers, or positive integers; that is to say, the numbers 1, 2, 3, 4, .... Positive integers have many properties, which will be found in treatises on the Theory of Integral Numbers; but at a very early stage in the development of mathematics it was found that the operations of Subtraction and Division could only be performed among them subject to inconvenient restrictions; and consequently, in elementary Arithmetic, classes of numbers are constructed such that the operations of subtraction and division can always be performed among them.

To obtain a class of numbers among which the operation of subtraction can be performed without restraint we construct the class of integers, which consists of the class of positive + integers (+ 1, +2, +3, ...) and of the class of negative integers (-1, -2, -3, ...) and the number 0.

To obtain a class of numbers among which the operations both of subtraction and of division can be performed freely, we construct the class of rational numbers. Symbols which denote members of this class are 1/2, 3, 0, - 1.5/7.

We have thus introduced three classes of numbers, (i) the signless integers, (ii) the integers, (iii) the rational numbers.

It is not part of the scheme of this work to discuss the construction of the class of integers or the logical foundations of the theory of rational numbers

The extension of the idea of number, which has just been described, was not effected without some opposition from the more conservative mathematicians. In the latter half of the eighteenth century, Maseres (1731-1824) and Frend (1757-1841) published works on Algebra, Trigonometry, etc., in which the use of negative numbers was disallowed, although Descartes had used them unrestrictedly more than a hundred years before.


 * Strictly speaking, a more appropriate epithet would be, not positive, but signless.

+ in the strict sense.

+ With the exception of division by the rational number 0.

Such a discussion, defining a rational number as an ordered number-pair of integers in a similar manner to that in which a complex number is defined in § 1.3 as an ordere i number-pair of real numbers, will be found in Hobson's Functions of a Real Variable, ss 1-12.

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